The lattice structure of preradicals III: operators

In this paper, we introduce the notion of essential extensions of fuzzy modules. We use these concepts to introduce the notion of injective hulls of fuzzy modules. It is known that every [Formula: see text]-module has an injective hull, where [Formula: see text] is a ring. We show that these corresponding results do not hold for fuzzy [Formula: see text]-modules, i.e. there exists a fuzzy [Formula: see text]-module that does not have an injective hull. Sufficient conditions are given for a fuzzy [Formula: see text]-module to have an injective hull.

In this paper, we introduce and study the properties of chain conditions on M-cyclic submodules defined as M-Noetherian and M-Artinian modules and rings. We prove that the property of being iso-M-Noetherian (iso-M-Artinian) is inherited by submodules, quotient modules, and finite sums as well as finite direct sums. Finally, we prove the Hilbert basis theorem for iso-M-Noetherian (iso-M-Artinian) and verify that the property of modules of being M-Noetherian (M-Artinian) and iso-M-Noetherian (iso-M-Artinian) are Morita invariant.

A module [Formula: see text] is called coseparable ([Formula: see text]-coseparable) if for every submodule [Formula: see text] of [Formula: see text] such that [Formula: see text] is finitely generated ([Formula: see text] is simple), there exists a direct summand [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ([Formula: see text]-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of [Formula: see text]-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ([Formula: see text]-coseparable).

Injective modules and semistar operations

We investigate blocks of the Category O for the Virasoro algebra over C. We demonstrate that the blocks have Kazhdan‐Lusztig theories and that the truncated blocks give rise to interesting Koszul algebras. The simple modules have BGG resolutions, and from this we compute the extensions between Verma modules and simple modules, and between pairs of simple modules.

In this paper R R denotes an associative ring with an identity, and all modules are unital left R R -modules. It is shown that the existence of a quasi-projective cover for each module implies that each module has a projective cover. By a similar technique the following statements are shown to be equivalent: 1. R R is semisimple and Artinian; 2. Every finitely generated module is quasi-projective; and 3. The direct sum of every pair of quasi-projective modules is quasi-projective. Direct sums of quasi-injective modules are also investigated.

We define the partition and $n$-point functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and for any pair of simple Heisenberg modules. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties for the Heisenberg and lattice vertex operator algebras and a continuous orbifolding of the rank two fermion vertex operator super algebra. We compute the genus two Heisenberg vector $n$-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity for these theories.

A right R-module M is virtually extending (or CIS) if every complement submodule of M is isomorphic to a direct summand of M, and M is called a virtually C2-module if every complement submodule of M which is isomorphic to a direct summand of M is itself a direct summand. The class of virtually extending modules (respectively, virtually C2-modules) is a strict and simultaneous generalization of extending modules (respectively, unifies extending modules and C2-modules): M is a semisimple module if and only if M is virtually semisimple and C2, and M is an extending module if and only if M is virtually extending and virtually C2. Furthermore, every virtually simple right R-module is injective if and only if R is a right V-ring and the class of virtually simple right R-modules coincides with the class of simple right R-modules. Among other results, we show that (1) if all cyclic sub-factors of a cyclic weakly co-Hopfian right R-module M are virtually extending, then M is a finite direct sum of uniform submodules; (2) every distributive virtually extending module over any Noetherian ring is a direct sum of uniform submodules; (3) over a right Noetherian ring, every virtually extending module satisfies the Schröder-Bernstein property; (4) being virtually extending (VC2) is a Morita invariant property; (4) if is a VC2-module where denotes the injective hull, then M is injective. Communicated by Scott Chapman

This note investigates torsion-free abelian groups G of finite rank which embed, as subgroups of finite index, in a finite direct sum C of subgroups of the additive group of rational numbers. Specifically, we examine the relationship between G and C when the index of G in C is minimal. Some properties of Warfield duality are developed and used (in the case that G is locally free) to relate our results to earlier ones by Burkhardt and Lady.

Let N be a normal subgroup of a group G. An N-module Q is called G-stable provided that Q is equivalent to the twist Q g of Q by g, for every g ∈ G. If the action of N on Q extends to an action of G on Q, then Q is obviously G-stable, but the converse need not hold. A famous conjecture in the modular representation theory of reductive algebraic groups G asserts that the (obviously G-stable) projective indecomposable modules (PIMs) Q for the Frobenius kernels G r (r ≥ 1) of G have a G-module structure. It is sometimes just as useful (for a general module Q) to know that a finite direct sum Q ⊕ n of Q has a compatible G-module structure. In this paper, this property is called numerical stability. In recent work (Parshall and Scott, Adv Math 226:2065–2088, 2011), the authors established numerical stability in the special case of PIMs. We provide in this paper a more general context for that result, working in the context of k-group schemes and a suitable version of G-stability, called strong G-stability. Among our results here is the determination of necessary and sufficient conditions for the existence of a compatible G-module structure on a strongly G-stable N-module, in the form of a cohomological obstruction which must be trivial precisely when the G-module structure exists. Our main result is achieved by giving an approach to killing the obstruction by tensoring with certain finite dimensional G/N-modules.

A module M is called (cofinitely) Rad-⊕-supplemented if every (cofinite) submodule of M has a Rad-supplement that is a direct summand of M. We prove that if M is a coatomic cofinitely Rad-⊕-supplemented module, then M is an irredundant sum of local direct summands. We show that the classes of cofinitely Rad-⊕-supplemented modules and Rad-⊕-supplemented modules are closed under finite direct sums. We also show that every direct summand of a weak duo (cofinitely) Rad-⊕-supplemented module is (cofinitely) Rad-⊕-supplemented.

On duality theory and AB5 ∗ modules

This paper investigates self-small abelian groups of finite torsion-free rank. We obtain a new characterization of infinite self-small groups. In addition, self-small groups of torsion-free rank 1 and their finite direct sums are discussed.

Classes of operators determined by ordinal indices

Harada (cf. 1983) introduced the lifting property for maximal submodules and the extending property for simple submodules, and Oshiro (cf. 1983b) definitely introduced lifting modules and extending modules. Since then, the method and results in their works urged not only the study on (quasi-)discrete and (quasi-)continuous modules but also the one on themselves. The reader is referred to Mohamed and Müller (1990) and Dung et al. (1994) for the research on lifting modules and extending modules. In this article, we study the following fundamental open problems: When is a direct sum ⨁IMi of lifting (extending) modules {Mi}I lifting (extending)? These problems are unsolved even in the case that the index set I is finite. Now, the purpose of this article is to study the problem concerned with finite direct sums of lifting modules.